to choose, one can always find a larger value of x for
which
is, and remains, less than –M for ever
larger values of x.
2. Geometry: In DESARGUES’S THEOREM, French
mathematician GIRARD DESARGUES (1591–1661) found
it convenient to regard parallel lines as intersecting at
some point of infinity. Thus in the theory of PROJECTIVE GEOMETRY, the notion of points “infinitely far
away” is given meaning and context.
3. Set theory: Italian astronomer and physicist
GALILEO GALILEI (1564–1642) observed that every
counting number can be matched with its square,
showing in some sense that the infinite set of counting
numbers is no more infinite than the subset of square
numbers:
In the 19th century, British algebraist AUGUSTUS DE
MORGAN (1806–71) and German mathematicians
JULIUS WILHELM RICHARD DEDEKIND (1831–1916) and
GEORG CANTOR (1845–1918) realized that this is a
common property of infinite sets, and that it is appropriate to use this property as the definition of what it
means for a set to be infinite:
A set is infinite if its elements can be matched,
without repetition, with the elements of a
proper subset of itself.
This definition has the advantage that it makes clear
what it means for a set to be finite.
A set is finite if it is not infinite.
(A comment should be made on this point. Although we
all have a clear intuitive understanding of what it means
for a set to be FINITE, it is not at all easy to provide a
direct mathematical description of this concept. However,
it is possible to show that no set of the form {1,2,3,…,n}
is infinite, that is, there is no means to match the elements of this set with the elements of a proper subset of
itself without producing repetition. This is done with an
INDUCTION argument on n. One can use this as a link
between this indirect approach and our intuitive understanding.) Cantor went further to develop an astounding
theory of CARDINALITY that shows, among other things,
that there are many different types of infinite sets, some
deserving of being called “more infinite” than others.
The notion of the infinite has been studied and used
since antiquity. ARCHIMEDES OF SYRACUSE (ca.287–212
B.C.E.) used the notion of an infinitely small quantity to
develop formulae for the areas and volumes of curved
figures and solids. (In some vague sense, one can view a
circle, for instance, as a regular polygon with infinitely
many sides, all infinitely short in length. It is better,
however, to view the circle as the LIMIT figure of a
sequence of regular polygons.) The geometer EUCLID
(ca.300–260 B.C.E.) proved that the set of PRIME numbers is infinite, and ZENO OF ELEA (ca. 490–425 B.C.E.)
contemplated the infinite in his studies of time, space,
and motion.
Although the scholars of Greek antiquity utilized
the infinite, they were wary of it. Euclid, for example,
went to great pains to phrase matters in a way that
never made mention of a quantity that was actually infinite. For instance, he proved that from any given finite
collection of primes, one can always construct one more
(rather than state that the set of prime numbers is infinite), and in his famous work THE ELEMENTS, he never
made mention of lines that continue indefinitely; he
only spoke of extending line segments further if needed.
(This subtle turn of phrase proved to be important to
GEORG FRIEDRICH BERNHARD RIEMANN with his 19thcentury invention of SPHERICAL GEOMETRY.) ARISTOTLE
(384–322 B.C.E.) argued that the “actual infinite” did
not exist, and that one can only argue in terms of
potentiality: given a finite part, one can always provide
more. This point of view held fast for almost two millennia. Cantor’s work on the actual infinite, inspired by
the beginning ideas of Dedekind and De Morgan, was
deemed revolutionary at its time.
inflection (inflexion) See CONCAVE UP/CONCAVE DOWN.
inflection point (point of inflection) A point on a
curve at which the tangent line to the curve changes
from rotating in one sense (clockwise or counterclockwise) to rotating in the opposite sense. The concavity
of the curve changes at such a point.
See also CONCAVE UP/CONCAVE DOWN.
1 2 3 4 5 6
1 4 9 16 25 36
...
...
b b b b b b
2
3
− x
x
270 inflection
which
is, and remains, less than –M for ever
larger values of x.
2. Geometry: In DESARGUES’S THEOREM, French
mathematician GIRARD DESARGUES (1591–1661) found
it convenient to regard parallel lines as intersecting at
some point of infinity. Thus in the theory of PROJECTIVE GEOMETRY, the notion of points “infinitely far
away” is given meaning and context.
3. Set theory: Italian astronomer and physicist
GALILEO GALILEI (1564–1642) observed that every
counting number can be matched with its square,
showing in some sense that the infinite set of counting
numbers is no more infinite than the subset of square
numbers:
In the 19th century, British algebraist AUGUSTUS DE
MORGAN (1806–71) and German mathematicians
JULIUS WILHELM RICHARD DEDEKIND (1831–1916) and
GEORG CANTOR (1845–1918) realized that this is a
common property of infinite sets, and that it is appropriate to use this property as the definition of what it
means for a set to be infinite:
A set is infinite if its elements can be matched,
without repetition, with the elements of a
proper subset of itself.
This definition has the advantage that it makes clear
what it means for a set to be finite.
A set is finite if it is not infinite.
(A comment should be made on this point. Although we
all have a clear intuitive understanding of what it means
for a set to be FINITE, it is not at all easy to provide a
direct mathematical description of this concept. However,
it is possible to show that no set of the form {1,2,3,…,n}
is infinite, that is, there is no means to match the elements of this set with the elements of a proper subset of
itself without producing repetition. This is done with an
INDUCTION argument on n. One can use this as a link
between this indirect approach and our intuitive understanding.) Cantor went further to develop an astounding
theory of CARDINALITY that shows, among other things,
that there are many different types of infinite sets, some
deserving of being called “more infinite” than others.
The notion of the infinite has been studied and used
since antiquity. ARCHIMEDES OF SYRACUSE (ca.287–212
B.C.E.) used the notion of an infinitely small quantity to
develop formulae for the areas and volumes of curved
figures and solids. (In some vague sense, one can view a
circle, for instance, as a regular polygon with infinitely
many sides, all infinitely short in length. It is better,
however, to view the circle as the LIMIT figure of a
sequence of regular polygons.) The geometer EUCLID
(ca.300–260 B.C.E.) proved that the set of PRIME numbers is infinite, and ZENO OF ELEA (ca. 490–425 B.C.E.)
contemplated the infinite in his studies of time, space,
and motion.
Although the scholars of Greek antiquity utilized
the infinite, they were wary of it. Euclid, for example,
went to great pains to phrase matters in a way that
never made mention of a quantity that was actually infinite. For instance, he proved that from any given finite
collection of primes, one can always construct one more
(rather than state that the set of prime numbers is infinite), and in his famous work THE ELEMENTS, he never
made mention of lines that continue indefinitely; he
only spoke of extending line segments further if needed.
(This subtle turn of phrase proved to be important to
GEORG FRIEDRICH BERNHARD RIEMANN with his 19thcentury invention of SPHERICAL GEOMETRY.) ARISTOTLE
(384–322 B.C.E.) argued that the “actual infinite” did
not exist, and that one can only argue in terms of
potentiality: given a finite part, one can always provide
more. This point of view held fast for almost two millennia. Cantor’s work on the actual infinite, inspired by
the beginning ideas of Dedekind and De Morgan, was
deemed revolutionary at its time.
inflection (inflexion) See CONCAVE UP/CONCAVE DOWN.
inflection point (point of inflection) A point on a
curve at which the tangent line to the curve changes
from rotating in one sense (clockwise or counterclockwise) to rotating in the opposite sense. The concavity
of the curve changes at such a point.
See also CONCAVE UP/CONCAVE DOWN.
1 2 3 4 5 6
1 4 9 16 25 36
...
...
b b b b b b
2
3
− x
x
270 inflection